On a nonlinear third-order equation

In this article, we study the fully non-linear third-order partial differential equation, namely the Gilson–Pickering equation. The $\left(1/{{G}^{'}}\right)$-expansion method, and the generalized exponential rational function method are used to construct various exact solitary wave solutions for a given equation. These methods are based on a homogeneous balance technique that provides an order for the estimation of a polynomial-type solution. In order to convert the governing equation into a nonlinear ordinary differential equation, a traveling wave transformation has been implemented. As a result, we have constructed a variety of solitary wave solutions, such as singular solutions, compound singular solutions, complex solutions, topological, and non-topological solutions. Besides, the 2D, 3D and contour surfaces are plotted for all obtained solutions by choosing appropriate parameter values.

An approximate theory is presented for post-stall transients in multistage axial compression systems. The theory leads to a set of three simultaneous nonlinear third-order partial differential equations for pressure rise, and average and disturbed values of flow coefficient, as functions of time and angle around the compressor. By a Galerkin procedure, angular dependence is averaged, and the equations become first order in time. These final equations are capable of describing the growth and possible decay of a rotating-stall cell during a compressor mass-flow transient. It is shown how rotating-stall-like and surgelike motions are coupled through these equations, and also how the instantaneous compressor pumping characteristic changes during the transient stall process.

It has been shown by Gupta and Padmanabhan that the radiation reaction force of the Abraham–Lorentz–Dirac equation can be obtained by a coordinate transformation from the inertial frame of an accelerating charged particle to that of the laboratory. We show that the problem may be formulated in a flat space of five dimensions, with five corresponding gauge fields in the framework of the classical version of a fully gauge covariant form of the Stueckelberg–Feynman–Schwinger covariant mechanics (the zero mode fields of the 0, 1, 2, 3 components correspond to the Maxwell fields). Without additional constraints, the particles and fields are not confined to their mass shells. We show that in the mass-shell limit, the generalized Lorentz force obtained by means of the retarded Green's functions for the five dimensional field equations provides the classical Abraham–Lorentz–Dirac radiation reaction terms (with renormalized mass and charge). We also obtain general coupled equations for the orbit and the off-shell dynamical mass during the evolution as well as an autonomous non-linear equation of third order for the off-shell mass. The theory does not admit radiation if the particle does not move off-shell. The structure of the equations implies that mass-shell deviation is bounded when the external field is removed.

Stability and boundedness in delay system of differential equations of third order

Solution of nonlinear partial differential equations from base equations

To design effective energy-absorbing shields made of modern composite materials (e.g., fibrous composite panels), it is necessary to use deformable materials whose important feature is the largest possible dissipation of impact energy. Rheological properties of these materials differ significantly from dissipative and elastic properties described by the Hooke and Kelvin linear models. Moreover, the range of material deformation at the place of impact is so wide that the above-mentioned models are not sufficient to properly describe the process of energy dissipation. The value of dissipated energy is also important for the determination of the lifespan of dissipative and elastic elements subjected to cyclic dynamic loading and made of modern structural materials (fatigue phenomena). The authors proposed a dynamic model in which the rheological properties of the material are described by two spring parameters c, c 0 and one parameter of viscous damping k 0 in the configuration used in the so-called rheological standard model (Zener model). In addition, the model introduces the element of dry friction described by the appropriate function h(x). In addition, it was assumed that the non-linear element present in this model describes the effect of the occurrence of dry friction hSign(dx/dt) in the tested material, the value of which h is not constant but depends to some extent on the level of material deformation (variable x) is described by the relevant function. In this way, a non-linear third-order equation was obtained describing the motion of the so-called degenerate system. The developed method was tested on a computer system.

In this paper, it is considered a quasi-linear strongly-damped wave equation defined by a non-linear partial differential equation of third order. The equation describes motions of viscoelastic solids. We study the conservation laws of this equation. By applying the multiplier method of Anco and Bluman to the equation, we find the multipliers. Consequently, we obtain a complete classification of conservation laws. Moreover, we use the Lie-group theory to analyse the symmetries of the equation. From the Lie symmetries, all the reductions are determined. Afterwards, we construct exact solutions with physical interest: travelling wave solutions.

The author consider initial-boundary value problem for a system of differential equations in partial derivatives of third order. We study questions of existence and uniqueness of classical solution of the problem, and ways for its evaluation. The introduction of a new desired function enables us to reduce the problem to equivalent nonlocal problem with integral condition for a system of integro-differential euqtions of hyperbolic type with a functional relation. We establish conditions for unique solvability of the nonlocal problem by means of the method of functional parameters, propose algorithms for solving of the equivalent problem and prove their convergence. As a result, there are obtained conditions for existence of unique classical solution of initial-boundary value problem for the considered system of differential equations of third order in terms of initial data.

In this study we investigate the existence and approximation of the periodic solutions for nonlinear system of differential equations of third order with boundary conditions. The numerical-analytic method has been used to study the periodic solutions of the ordinary differential equations that were introduced by Samoilenko. Also these investigation lead us to the improving and extending the above method and the results of Butris.

In this paper, the authors give sufficient conditions for the boundedness and global asymptotic stability of solutions to certain nonlinear multi‐delay functional differential equations of the third order. The technique of proof involves defining an appropriate Lyapunov‐Krasovskii functional and applying LaSalle's invariance principle. An example is included to illustrate the results. Copyright © 2014 John Wiley & Sons, Ltd.

This chapter is concerned with the study of oscillatory and asymptotic behaviour of nonoscillatory solutions of nonhomogeneous third-order differential equations of the form $$x^{\prime\prime\prime} + a(t) x^{\prime\prime} + b(t)x^{\prime} + c(t) x = f \bigl(t,x,x^{\prime},x^{\prime\prime}\bigr), $$ where a, b and c∈C([σ,∞),R) and f:[σ,∞)×R 3→R, σ∈R. z-type oscillation criteria has been used in this chapter to study the nonoscillation of solutions of the considered equation. As an application, some sufficient conditions have been given for the nonoscillation of different mathematical models in Engineering.

This paper gives some sufficient conditions for every solution of delay differential equation $$ \multline \buildrel \ldots \over x (t) +f(t,x(t),x(t-r),\dot{x}(t),\dot{x}(t-r),\ddot{x}(t),\ddot{x}(t-r)) \\ +b(t)g(x(t-r),\dot{x}(t-r)) +c(t)h(x(t)) \\ =p(t,x(t),x(t-r),\dot{x}(t),\dot{x}(t-r),\ddot{x}(t)) \endmultline $$ to be bounded.

: Complex physical phenomena involving chemically reacting systems or the transport of heat or fluids are often modeled by coupled systems of time-dependent, nonlinear partial differential equations. The difficulties in understanding the stability of the differential equation systems and in designing efficient, accurate numerical methods for their solution are widely recognized and were the focus of this research. We have worked on four general aspects of the analysis and numerical approximation of systems of partial differential equations. These areas of research are: (1) modeling aspects and stability analysis for nonlinear time-dependent partial differential equations; (2) use and analysis of finite element or finite difference methods to discretize coupled systems of nonlinear differential equations; (3) development of adaptive or local grid refinement capabilities to resolve local phenomena in large-scale applications; and (4) development of data structures, preconditioners, and efficient solution algorithms for large-scale problems on new computer architectures. Emphasis has been placed upon multiphase or multicomponent, transport-dominated flow processes with dynamic local phenomena. The research also involved a mix of analysis, algorithm development, and large-scale computation using newer computer architectures.

The equation of slow diffusion with singular boundary data is considered. An estimate of all weak solutions of such a problem is obtained, provided that the boundary regime is localized. The comparative analysis of the results obtained by the method of energy estimates and the barrier technique for the equation of porous medium is presented.

The equation of slow diffusion with singular boundary data is considered. An estimate of all weak solutions of such a problem is obtained, provided that the boundary regime is localized. The comparative analysis of the results obtained by the method of energy estimates and the barrier technique for the equation of porous medium is presented.